§2.1.1 Stress

According to the theory of elasticity, external forces acting on a solid-state body produce internal forces between the portions of the body and cause deformation. If the external forces do not exceed a certain limit, the deformation disappears once the forces are removed. To describe the internal forces, the stress tensor is introduced. Mathematically, stress is a second rank tensor, which has nine components as shown by the matrix

(2.1.1)

where the three diagonal components are referred to as normal stresses and the six off-diagonal components are called shearing stresses.

To illustrate the definition of the components of stress tensor, let us examine an elemental cube inside the body as shown in Fig. 2.1.1. The six faces of the cube are denoted as x, , y, , z, , according to the normal of the faces. (Note: a bar over a letter indicates a negative sign for the letter.)

In the figure, Tij, the component of a stress tensor, is defined as the force in a specific direction j (the second subscript) on a unit area on a specific surface i (the first subscript) of the elemental cube. For examples, TXX in Fig. 2.1.1 is the normal force per unit area of the x-plane, TXY is the force in the y-direction applied on a unit area of the x-plane and TXZ is the force in the z-direction per unit area of x-plane, and so forth.

The signs of the tensor components are determined by the direction of the force relative to the normal of the plane. For example, for the x-plane, the normal stress component caused by a force in the x-direction is defined as positive but that caused by a force in the -direction is defined as negative. The stress component caused by the tangential forces on the -plane and in the y- and z-directions are defined as positive while those by tangential forces in the - and -directions are defined as negative. Furthermore, for the plane, the stress component caused by a force in the -direction is positive but that caused by a force in the y-direction is negative. Similarly, the stress components caused by the tangential forces in the - and -directions on the -plane are defined as positive and those by the forces in the x- and z- directions on the -plane are defined as negative.

According to the condition of force balance, the TXX in two opposite parallel planes (x- and x -planes) should be equal in quantity and sign, and the same is true for the TYY and the TZZ. Also from the condition of torque balance, we have

(2.1.2)

This means that the matrix of stress tensor T is symmetric and has only six independent components. Therefore, they are often denoted by a simplified notation system

(2.1.3)

Therefore, equation (2.1.1) can be written as

§2.1.2 Strain

With stresses, deformation is produced inside the material. For easy to understand, let us first look at the deformation of a one-dimensional material as shown in fig. 2.1.2. If the displacement of an original position x is u(x) and the displacement of an original position x+Δx is u(x+Δx), the strain (the relative elongation of the material) in the one-dimensional material has only one component,

For a three dimensional material, the deformation of the material is schematically shown in Fig. 2.1.3. The displacement components for the point r (x,y,z) are u(x,y,z), v(x,y,z) and w(x,y,z) in the x-, y- and z-directions, respectively, and the displacement components for the point r′ (x+Δx, y+Δy, z+Δz) are u+Δu, v+Δv and w+Δw, respectively. The three components are all functions of x, y and z.

Therefore, the incremental displacement between point r (x, y, z) and point r′ (x+Δx, y+Δy, z+Δz) can be expressed as

(2.1.4)

For a solid-state body rotating around the origin O of a coordinate system with an angular velocity , the velocity of the end of the vector is . According to equations , we have .

Therefore, an angular displacement can be expressed as

(2.1.5)

where

Therefore, the last term on the right side of Eq. (2.1.4) is

If no rotational movement for the solid-state material is allowed, the last term on the right side of Eq. (2.1.4) is zero and the equation can be written as

(2.1.6)

The 3 by 3 matrix in the equation is referred to as a strain tensor in solid-state physics. The three diagonal components in the matrix are called the normal strain components of the strain tensor

(2.1.7)

It is quite clear that the three normal components of strain tensor shown in Eq. (2.1.7) are the relative elongations along the three coordinate axes. The six off-diagonal components are referred to as the shearing strain components of the strain tensor

(2.1.8)

Therefore, Equation (2.1.6) is written as

(2.1.9)

where (e) represents the strain tensor, a tensor of the second rank. As the strain tensor (e) is a symmetrical tensor with only six independent components, simplified notations can be used

Thus, we have

It has been found that the three shearing strain components are related to the angular distortion of the material. To verified these results, we consider the distortion of angle ∠APB, a right angle included by the elemental sections of PA=dx and PB=dy in the XY plane as shown in Fig. 2.1.4. Due to a deformation, the original positions A, P and B move to A′, P′, and B′, respectively. If u(x, y) and v(x, y) are the displacements in the x- and y-directions for point P(x,y), respectively, the displacement of point A in the y-direction is

And the displacement of point B in the x-direction is

The...